The title is a little nonsensical (so feel free to edit as you see fit)
Logb(n) where n is a power of b produces a rational number; for example; Log2(8) = 3
But, Log2(3) = 1.5849625007211561814537389439478165087598144076924810604557526545...
Thus, Log2(3) produces an irrational number.
Is the same true for any Logb(n) where n is not a power of b, or are there cases where this produces a rational number?
The logarithm function is the inverse of the exponential function. So $\log_b x = a \iff x = b^a$.
Thus, if $\log_b x = a$ is rational, then we can write $a = \frac{m}{n}$ where $m$ and $n$ are integers, and so $x = b^\frac{m}{n}$. But by applying power rules we can see that this is equivalent to $x = (b^m)^\frac{1}{n} = \sqrt[n]{b^m}$, i.e. $x$ is the $n$th root of $b^m$.
All of these steps are reversible, meaning that if $x$ is a rational power of $b$, then $\log_b x$ is a rational number and vice versa.